The discussion centers on proving that for a finite group G with a normal subgroup H of finite index m, the element a raised to the power of m (a^m) belongs to H for all elements a in G. The canonical map φ: G → G/H, defined by g ↦ gH, is suggested as a tool to analyze the behavior of a^m. Additionally, it is clarified that the statement "order of a group equals the order of an element" is only true for cyclic groups where the element is a generator.
PREREQUISITESThis discussion is beneficial for students of abstract algebra, particularly those studying group theory, as well as educators and researchers looking to deepen their understanding of normal subgroups and cyclic groups.
]. Show that a^m\in H for all a\in G.